Compact Symmetric Spaces, Triangular Factorization, and Poisson Geometry

A. Caine

Journal of Lie theory, Tome 18 (2008) no. 2, pp. 273-294.

Résumé

\def\C{{\Bbb C}} \def\g{{\frak g}} \def\h{{\frak h}} \def\n{{\frak n}} \def\u{{\frak u}} Let $X$ be a simply connected compact Riemannian symmetric space, let $U$ be the universal covering group of the identity component of the isometry group of $X$, and let $\g$ denote the complexification of the Lie algebra of $U$, $\g=\u^\C$. Each $\u$-compatible triangular decomposition $\g=\n_- + \h + \n_+$ determines a Poisson Lie group structure $\pi_U$ on $U$. The Evens-Lu construction produces a $(U,\pi_U)$-homogeneous Poisson structure on $X$. By choosing the basepoint in $X$ appropriately, $X$ is presented as $U/K$ where $K$ is the fixed point set of an involution which stabilizes the triangular decomposition of $\g$. With this presentation, a connection is established between the symplectic foliation of the Evens-Lu Poisson structure and the Birkhoff decomposition of $U/K$. This is done through reinterpretation of results of Pickrell. Each symplectic leaf admits a natural torus action. It is shown that the action is Hamiltonian and the momentum map is computed using triangular factorization. Finally, local formulas for the Evens-Lu Poisson structure are displayed in several examples.

Détail

Classification : 53D17, 53D20, 53C35
Mots-clés : Homogeneous poisson structures, symmetric spaces, momentum map